“Journey Through Genius” by William Dunham

A history of mathematics told through a dozen “great theorems,” from Hippocrates to Cantor. I’m not sure, after reading, who precisely is the intended audience: the amount of actual mathematics seemed a bit heavy for a layperson. I was somewhat skeptical of the text when I started, but my impression improved after getting further in. Dunham manages to weave threads throughout several chapters, giving a coherence to what otherwise could have been disjointed stories separated by centuries. I also found several of his remarks to be quite intriguing. On the specialness of mathematics, he observes that even though modern mathematicians consider much of Euclid questionable in one way or another,

none of his 465 theorems is false … all have withstood the test of time. … [One might] compare Euclid’s record with that of Greek astronomers or chemists or physicians.

Later, I enjoyed very much Dunham’s discussion of the trend (starting in the 19th century) towards mathematics not necessarily tied to the study of the physical world, and its parallels in the rise of non-representational art.

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“Best American Science Writing 2011″

Edited by Rebecca and Floyd Skloot. Of the handful of books in this series I’ve read, this was definitely among the best: very few selections suffered from absurd fanboyism, obvious inanity, or just bad writing, and several pieces were absolutely excellent. I was surprised there were not articles by Elizabeth Kolbert, just because she’s consistently among the very best science writers out there. The decision to include several shorter pieces drawn from the New York Times was interesting.

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Pedantry

Mathematicians are notoriously pedantic.  Two real-life stories, which serve as my replacements for the terrible black sheep joke:

  • I arrive at a conference dormitory a bit early, just as distribution of keys is beginning.  A staff member apologizes that she only has the box with assignments for people whose last names begin with a letter from M to Z, but that the A to Ls should be arriving shortly.  As I pass the time making small talk, a friend arrives and inquires why we’re just standing around.  I respond, “Our names are in the wrong half of the alphabet.”  Another attendee overhears this remark and steps over.  “Well, technically M through Z is slightly more than half of the alphabet.”
  • The professor of a class I’ve been taking gives a talk in our seminar.  For much of the talk, he carries his mug in hand while he works at the blackboard, sipping occasionally.  In office hours later that week, I commend him on his ability to hold a coffee cup while lecturing.  He responds, “I don’t drink coffee, only water.”
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A few labor things

  • The one-day strikes at fast food and other retailers are wonderful and make me optimistic about the world; in DC, they’ve already (by a stroke of luck) resulted in a union for some workers and have been (along with the Walmart protests) a major part of the recent minimum-wage increase there.
  • The push by college football players to form a union is amazing.
  • Locally, there will be a minimum wage increase in Minnesota this year.  The only question is where it falls between the House’s proposed $9.50 and the (absurdly low) Senate proposal of $7.75.  I suggest local residents spend some quality time on the phone with your state senators pointing out how worthless a raise to $7.75 would be.
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Alan Alda

Alan Alda is absolutely the sexiest 70+ year old out there.  He’s also a fabulous communicator of science — perhaps, along with Neil deGrasse Tyson, the best of the present generation.

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1 + 2 + 3 + 4 + …

There’s a video that has gone viral and concerns itself with the equation

1 + 2 + 3 + 4 + \ldots = - \dfrac{1}{12}.

A couple of students from my Calculus 2 class last semester sent me the following e-mail, which makes me smile for several reasons:

We are rather skeptical, but do not really understand explanations of the “proofs” others have provided. Since we consider you the expert on convergence and divergence we were hoping you knew what the answer actually is. Is this a result that’s accepted by the mathematical community? Or are we just being lied to by the internet?

I think Calculus 2 (which includes a fairly detailed coverage of Taylor series) is a great moment to be asking this question; here was my response:

The very short answer to your question is, no, it is not really true that the sum of the positive integers is equal to −1/12.  But there is an interesting piece of mathematics that lies behind that equation, and it’s not quite the total nonsense it appears on first glance.  I recommend the following blog post by a professor at Utah:

Here is my supporting commentary to go along with it.  The mathematical context called is the field called “complex analysis” — it’s essentially what you get if you do calculus, but you use the complex numbers as your basic set of numbers instead of the real numbers.  Many things work out exactly the same; for example, the definition of the derivative is exactly the same, except that the little h going to 0 is now a complex number instead of a real number.  And one can still build up all the machinery of Taylor series in this context.  Now, we’ve seen that sometimes Taylor series converge only in some cases.  For example, we know
1+x+x^2+x^3+\ldots = \dfrac{1}{1 - x}
but only for certain values of x (namely |x| < 1), even though the function on the right is defined in other situations (for any x other than 1).  So, the equation you’re asking about is similar to
1+2+4+8+16+\ldots = -1

with the “proof” that you put x=2 into that formula for geometric series above.  This is not right (we’re outside the interval of convergence, and the series on the left diverges), but it’s possible to understand what is meant by the equation.  The 1+2+3+4+… example is similar, but with a more sophisticated series.

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Resolution

Q: What do you call it when in early January you use an exact sequence to describe the structure of a module?

Continue reading

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